I’m no expert on this corner of history, so take this for what it’s worth, but military issued axe guns, and guns with bayonets existed at the overlapping times.

If I recall correctly, bayonets were an outgrowth of pike and shot warfare, where eventually the duties of the pike were taken over by the bayonet in infantry use in open field warfare.

Axe guns were used by cavalry, which makes sense as a short and swingable weapon; axe guns were issued to naval troops as boarding weapons, which makes sensible for the close quarter combat compared to a bayonet on a musket.